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GEOMETRIC PATTERN OF POWERS OF THREE VIA SIERPINSKI TRIANGULAR NUMBERS

Year 2024, Volume: 25 Issue: 3, 485 - 489, 30.09.2024
https://doi.org/10.18038/estubtda.1491452

Abstract

In this study, we describe a new number sequence based on integers arranged in a fractal like structure to visualize powers of three. Moreover, we associate these new numbers with triangular numbers

References

  • [1] Bruckman P, Dence JB, Dence TP, Young J. Series of reciprocal triangular numbers. The College Math J 2013; 44(3): 177-184.
  • [2] Caglayan G. Covering a triangular number with pentagonal numbers. Math Intelligencer 2020; 42: 55.
  • [3] Cagman A. Explicit solutions of powers of three as sums of three Pell numbers based on Baker’s type inequalities. Turkish Journal of Inequalities 2021; 5(1): 93-103.
  • [4] Coons M, Winning HH. Powers of two modulo powers of three. Journal of Integer Sequences 2015; 18: article 15.6.1.
  • [5] Charles FM. Proof without words: Square triangular sums. Mathematics Magazine 2019; 92(4): 269.
  • [6] Edgar T. Proof without words: A recursion for triangular numbers and more. Mathematics Magazine 2017; 90(2): 124-125.
  • [7] Edgar T. Visual triangular number identities from positional number systems. The College Math J 2021; 52(2): 133-136.
  • [8] Hopkins B. Proof without words: Products of odd squares and triangular numbers. Mathematics Magazine 2018; 91(1): 42.
  • [9] Jones MA. Proof without words: The square of a balancing number is a triangular number. The College Math J 2012; 43(3): 212.
  • [10] Leach CD. Proof without words: Powers of three and triangular numbers. The College Math J 2016; 47: 120.
  • [11] Matthew JH, Jones MA. Proof without words: Nonnegative integer solutions and triangular numbers. Mathematics Magazine 2002; 75(5): 388.
  • [12] Nelsen RB. Proof without words: Squares of triangular numbers. Mathematics Magazine 1990; 63(3): 178.
  • [13] Nelsen RB. Proofs without words II. Classroom Resource Materials. Mathematical Association of America, pp 108, 2000.
  • [14] Plaza Á. Proof without words: Sum of triangular numbers. Mathematics Magazine 2016; 89(1): 36-37.
  • [15] Stephen LS. Proof without words: Alternating sum of squares = triangular number. Mathematics Magazine 1992; 65(2): 90.
  • [16] Tiebekabe P, Diouf I. Powers of three as difference of two Fibonacci numbers. JP Journal of Algebra, Number Theory and Applications 2021; 49(2): 185-196.
  • [17] Unal H. A visual proof for the sum of the first n triangular numbers. Math Intelligencer 2010; 32: 6-7.
  • [18] Zerger MJ. Proof without words: Sums of triangular numbers. Mathematics Magazine 1990; 63(5): 314.
Year 2024, Volume: 25 Issue: 3, 485 - 489, 30.09.2024
https://doi.org/10.18038/estubtda.1491452

Abstract

References

  • [1] Bruckman P, Dence JB, Dence TP, Young J. Series of reciprocal triangular numbers. The College Math J 2013; 44(3): 177-184.
  • [2] Caglayan G. Covering a triangular number with pentagonal numbers. Math Intelligencer 2020; 42: 55.
  • [3] Cagman A. Explicit solutions of powers of three as sums of three Pell numbers based on Baker’s type inequalities. Turkish Journal of Inequalities 2021; 5(1): 93-103.
  • [4] Coons M, Winning HH. Powers of two modulo powers of three. Journal of Integer Sequences 2015; 18: article 15.6.1.
  • [5] Charles FM. Proof without words: Square triangular sums. Mathematics Magazine 2019; 92(4): 269.
  • [6] Edgar T. Proof without words: A recursion for triangular numbers and more. Mathematics Magazine 2017; 90(2): 124-125.
  • [7] Edgar T. Visual triangular number identities from positional number systems. The College Math J 2021; 52(2): 133-136.
  • [8] Hopkins B. Proof without words: Products of odd squares and triangular numbers. Mathematics Magazine 2018; 91(1): 42.
  • [9] Jones MA. Proof without words: The square of a balancing number is a triangular number. The College Math J 2012; 43(3): 212.
  • [10] Leach CD. Proof without words: Powers of three and triangular numbers. The College Math J 2016; 47: 120.
  • [11] Matthew JH, Jones MA. Proof without words: Nonnegative integer solutions and triangular numbers. Mathematics Magazine 2002; 75(5): 388.
  • [12] Nelsen RB. Proof without words: Squares of triangular numbers. Mathematics Magazine 1990; 63(3): 178.
  • [13] Nelsen RB. Proofs without words II. Classroom Resource Materials. Mathematical Association of America, pp 108, 2000.
  • [14] Plaza Á. Proof without words: Sum of triangular numbers. Mathematics Magazine 2016; 89(1): 36-37.
  • [15] Stephen LS. Proof without words: Alternating sum of squares = triangular number. Mathematics Magazine 1992; 65(2): 90.
  • [16] Tiebekabe P, Diouf I. Powers of three as difference of two Fibonacci numbers. JP Journal of Algebra, Number Theory and Applications 2021; 49(2): 185-196.
  • [17] Unal H. A visual proof for the sum of the first n triangular numbers. Math Intelligencer 2010; 32: 6-7.
  • [18] Zerger MJ. Proof without words: Sums of triangular numbers. Mathematics Magazine 1990; 63(5): 314.
There are 18 citations in total.

Details

Primary Language English
Subjects Algebraic and Differential Geometry
Journal Section Articles
Authors

Temel Ermiş 0000-0003-4430-5271

Publication Date September 30, 2024
Submission Date May 28, 2024
Acceptance Date July 30, 2024
Published in Issue Year 2024 Volume: 25 Issue: 3

Cite

AMA Ermiş T. GEOMETRIC PATTERN OF POWERS OF THREE VIA SIERPINSKI TRIANGULAR NUMBERS. Eskişehir Technical University Journal of Science and Technology A - Applied Sciences and Engineering. September 2024;25(3):485-489. doi:10.18038/estubtda.1491452